SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
MATT KERR
Abstract. These largely expository notes are based on a minicourse given by the author at the 2010 ICTP Summer School on
Hodge Theory.
Introduction
In algebraic geometry there is a plethora of objects which turn out
by big theorems to be algebraic, but which are defined analytically:
• projective varieties, and functions and forms on them (by Chow’s
theorem or GAGA [Serre1956]);
• Hodge-loci, and zero-loci of normal functions (work of CattaniDeligne-Kaplan [CDK1995], and Brosnan-Pearlstein [BP2009]);
• complex tori with a polarization (using theta functions, or the
embedding theorem [Kodaira1954]);
• Hodge classes (if a certain $1,000,000 problem could be solved);
and of concern to us presently:
• modular (locally symmetric) varieties,
which can be thought of as the Γ \D ’s for the period maps of certain
special VHS’s. The fact that they are algebraic is the Baily-Borel
theorem [BB1966].
What one does not know in the Hodge/zero locus setting above is
the field of definition — a question related to the existence of BlochBeilinson filtrations, which are discussed in M. Green’s course in this
volume. For certain cleverly constructed unions of modular varieties,
called Shimura varieties, one actually knows the minimal (i.e. reflex) field of definition, and also quite a bit about the interplay between “upstairs” and “downstairs” (in Ď resp. Γ \D ) fields of definition
of subvarieties. My interest in the subject stems from investigating
Mumford-Tate domains of Hodge structures, where for example the reflex fields can still be defined even though the Γ \D ’s are not algebraic
varieties in general (cf. §5, and [GGK2010]). Accordingly, I have tried
to pack as many Hodge-theoretic punchlines into the exposition below
as possible.
1
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
2
Of course, Shimura varieties are of central importance from another
point of view, that of the Langlands program. For instance, they
provide a major test case for the conjecture, generalizing ShimuraTaniyama, that all motivic L-functions (arising from Galois representations on étale cohomology of varieties over number fields) are automorphic, i.e. arise from automorphic forms (or more precisely from
Hecke eigenforms of adelic algebraic groups). The modern theory is
largely due to Deligne, Langlands and Shimura (with crucial details by
Shih, Milne, and Borovoi), though many others are implicated in the
huge amount of underlying mathematics: e.g.
• complex multiplication for abelian varieties (Shimura, Taniyama,
Weil);
• algebraic groups (Borel, Chevalley, Harish-Chandra);
• class field theory (Artin, Chevalley, Weil; Hensel for p-adics);
• modular varieties (Hilbert, Hecke, Siegel) and their compactifications (Baily, Borel, Satake, Serre, Mumford).
It seems that much of the impetus, historically, for the study of locally
symmetric varieties can be credited to Hilbert’s 12th problem generalizing Kronecker’s Jugendtraum. Its goal was the construction of abelian
extensions (i.e. algebraic extensions with abelian Galois groups) of certain number fields by means of special values of abelian functions in
several variables, and it directly underlay the work of Hilbert and his
students on modular varieties and the theory of CM.
What follows is based on the course I gave in Trieste, and (though
otherwise self-contained) makes free use of the notes in this volume by
E. Cattani (variations of Hodge structure), J. Carlson (period maps
and period domains), and P. Griffiths (Mumford-Tate groups). It was
a pleasure to speak at such a large and successful summer school, and
I heartily thank the organizers for the invitation to lecture there. A
brief outline follows:
(1)
(2)
(3)
(4)
(5)
Hermitian symmetric domains — D
Locally symmetric varieties — Γ \D
The theory of complex multiplication
Shimura varieties — qi (Γi \D )
The field of definition
1. Hermitian symmetric domains
A. Algebraic groups and their properties.
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
3
Definition 1.1. An algebraic group G over a field k (of characteristic
zero) is a smooth algebraic variety G together with morphisms
•: G×G→G
(·)−1 : G → G
(multiplication)
(inversion)
defined over k and an element
e ∈ G(k)
(identity) ,
subject to rules which make G(L) into a group for each L/k. Here G(L)
denotes the L-rational points of G, i.e. the morphisms SpecL → G.
In particular, G(R) and G(C) have the structure of real resp. complex
Lie groups.
Exercise 1.2. Write out these rules as commutative diagrams.
Example 1.3. As an algebraic variety, the multiplicative group
(GL1 ∼
=) Gm := {XY = 1} ⊂ A2 ,
with Gm (k) = k ∗ .
We review the definitions of some basic properties,1 starting with
• G connected ⇐⇒ Gk̄ irreducible,
where the subscript denotes the extension of scalars: for L/k, GL :=
G ×Speck SpecL. There are two fundamental building blocks for algebraic groups, simple groups and tori:
• G simple ⇐⇒ G nonabelian, with no normal connected subgroups 6= {e}, G.
Example 1.4. (a) k = C: SLn [Cartan type A], SOn [types B, D],
Spn [type C], and the exceptional groups of types E6 , E7 , E8 , F4 , G2 .
(b) k = R: have to worry about real forms (of these groups) which
can be isomorphic over C but not over R.
(c) k = Q: Q-simple does not imply R-simple; in other words, all
hell breaks loose.
• G (algebraic) torus ⇐⇒ Gk̄ ∼
= Gm × · · · × Gm
Example 1.5. (a) k = C: the algebraic tori are all of the form (C∗ )×n .
(b) k = Q: given a number field E, the “Weil restriction” or “restriction of scalars” G = ResE/Q Gm is a torus of dimension [E : Q] with
the property that G(Q) ∼
= E ∗ , and more generally G(k) ∼
= E ∗ ⊗Q k. If
k ⊇ E then G splits, i.e.
G(k) ∼
= (k ∗ )[E:Q] ,
1for
the analogous definitions for real and complex Lie groups, see [Rotger2005]
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
4
with the factors corresponding to the distinct embeddings of E into k.
(c) k ⊂ C arbitrary: inside GL2 , one has tori U ⊂ S ⊃ Gm with
k-rational points
a b a2 + b2 = 1
U(k) =
−b a a, b ∈ k
a b a2 + b2 6= 0
S(k) =
−b a a, b ∈ k
α 0 ∗
Gm (k) =
a∈k .
0 α In particular, their complex points take the form
C∗ z

/
/
C∗ × C∗ o
(z, z̄) ; (α, α) o
? _ C∗
α
by considering the eigenvalues of matrices. Writing S 1 or U1 for the
unit circle in C∗ , the real points of these groups are U1 ⊂ C∗ ⊃ R∗ , the
smaller two of which exhibit distinct real forms of Gm,C . The map
 : U → Gm
sending
a b
a + bi
0
7→
−b a
0
a + bi
is an isomorphism on the complex points but does not respect real
points (hence is not defined over R).
Next we put the building blocks together:
• G semisimple ⇐⇒ G an almost-direct product of simple
subgroups,
i.e. the morphism from the direct product to G is an isogeny (has
zero-dimensional kernel). More generally,
• G reductive ⇐⇒ G an almost-direct product of simple groups
and tori,
which turns out to be equivalent to the complete reducibility of G’s
finite-dimensional linear representations.
Example 1.6. One finite-dimensional representation is the adjoint
map
Ad
G −→
GL(g)
,
g 7−→ {X 7→ gXg −1 }
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
5
where g = Lie(G) = Te G and we are taking the differential (at e) of
Ψg ∈ Aut(G), i.e. conjugation by g.
We conclude the review with a bit of structure theory. For semisimple
groups:
• G adjoint ⇐⇒ Ad is injective; and
• G simply connected ⇐⇒ any isogeny G0 G with G0
connected is an isomorphism.
Basically, the center Z := Z(G) is zero-dimensional, hence has finitely
many points over k̄; we can also say that Z is finite. Adjointness is
equivalent to its triviality; whereas in the simply connected case Z is
as large as possible (with the given Lie algebra).
For reductive groups, we have short-exact sequences
Z0
(
Z
Gder
/
G
Ad
'
/
Gad
T
with Gder := [G, G] and Gad := Ad(G) both semisimple (and the latter
adjoint), Z 0 := Z ∩ Gder finite, and T a torus. Clearly, G is semisimple
if and only if the “maximal abelian quotient” T is trivial.
Finally, let G be a reductive real algebraic group,
θ: G→G
an involution.
Definition 1.7. θ is Cartan ⇐⇒
{g ∈ G(C) | g = θ(ḡ)} =: G(θ) (R) is compact.
Equivalently, θ = ΨC for C ∈ G(R) with
• C 2 ∈ Z(R) and
• G ,→ Aut(V, Q) for some symmetric bilinear form Q
¯ > 0 on VC .
satisfying Q(·, C (·))
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
6
Of course, to a Hodge theorist this last condition suggests polarizations and the Weil operator.
Cartan involutions always exist, and
G(R) compact ⇐⇒ θ = id.
B. Three characterizations of Hermitian symmetric domains.
I. Hermitian symmetric space of noncompact type. This is the “intrinsic analytic” characterization. The basic object is (X, g), a connected
complex manifold with Hermitian metric, or equivalently a Riemannian manifold with integrable almost complex structure such that J acts
by isometries. The real Lie group Is(X, g) of holomorphic isometries
must
• act transitively on X, and
• contain (for each p ∈ X) symmetries sp : X → X with s2p =
idX and p as isolated fixed point; moreover,
• the identity connected component Is(X, g)+ must be a semisimple adjoint noncompact (real Lie) group.
The noncompactness means that (a) the Cartan involution projects to
the identity in no factor and (b) X has negative sectional curvatures.
II. Bounded symmetric domain. Next we come to the “extrinsic analytic” approach, with X a connected open subset of Cn with compact
closure such that the (real Lie) group Hol(X) of holomorphic automorphisms
• acts transitively, and
• contains symmetries sp as above.
The Bergman metric makes X into a noncompact Hermitian space, and
the Satake (or Harish-Chandra) embedding does the converse job. For
further discussion of the equivalence between I and II, see [Milne2005,
sec. 1].
III. Circle conjugacy class. Finally we have the “algebraic” version,
which will be crucial for any field of definition questions. Moreover,
up to a square root, this is the definition that Mumford-Tate domains
generalize as we shall see later.
Let G be a real adjoint (semisimple) algebraic group. We take X to
be the orbit, under conjugation by G(R)+ , of a homomorphism
φ: U→G
of algebraic groups defined over R subject to the constraints:
• only z, 1, and z −1 appear as eigenvalues in the representation
Ad ◦ φ on Lie(G)C ;
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
7
• θ := Ψφ(−1) is Cartan; and
• φ(−1) doesn’t project to the identity in any simple factor of G.
The points of X are of the form gφg −1 for g ∈ G(R)+ , and we shall
think of it as “a connected component of the conjugacy class of a circle
in G(R)”. Obviously the definition is independent of the choice of φ in
a fixed conjugacy class.
Under the equivalence of the three characterizations,
Is(X, g)+ = Hol(X)+ = G(R)+ .
If, in any of these groups, Kp denotes the stabilizer of a point p ∈ X,
then
∼
=
G(R)+ Kp −→ X.
We now sketch the proof of the equivalence of the algebraic and
(intrinsic) analytic versions.
From (III) to (I). Let p denote a point of X given by a circle homomorphism φ. Since the centralizer
K := ZG(R)+ (φ)
belongs to G(θ) (R),
(a) kC := Lie(KC ) is the 1-eigenspace of Adφ(z) in gC , and
(b) K is compact (in fact, maximally so).
By (a), we have a decomposition
gC = kC ⊕ p− ⊕ p+
into Adφ(z)-eigenspaces with eigenvalues 1, z, z −1 . Identifying p− ∼
=
gR /k puts a complex structure on Tφ X for which d(Ψφ(z) ) is multiplication by z. Using G(R)+ to translate J := d(Ψφ(i) ) to all of T X yields
an almost complex structure.
One way to see this is integrable, making X into a (connected) complex manifold, is as follows: define the compact dual X̌ to be the
AdG(C)-translates of the flag
1
F = p+
F 0 = p+ ⊕kC
on gC . We have
X̌ ∼
= G(C)/ P ∼
= G(θ) (R) K
for P a parabolic subgroup with Lie(P ) = kC ⊕ p+ , which exhibits X̌
as a C-manifold and as compact. (It is in fact projective.) The obvious
map from decompositions to flags yields an injection X ,→ X̌ which is
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
8
an isomorphism on tangent spaces, exhibiting X as an (analytic) open
subset of X̌.
Now by (b), there exists a K-invariant symmetric and positivedefinite bilinear form on Tφ X. Translating this around yields a G(R)+ invariant Riemannian metric g on X. Since J ∈ K, g commutes with
(translates of) J and is thereby Hermitian. The symmetry at φ is given
by sφ := Ψφ(−1) (acting on X). Since Ψφ(−1) (as an element of Aut(G))
is Cartan and doesn’t project to e in any factor, G is noncompact. From (I) to (III). Since Is(X, g)+ is adjoint and semisimple, by [Borel1991,
Thm. 7.9] it is G(R)+ for some algebraic group G ⊂ GL(Lie(Is(X, g)+ )).
(Note that this can only make sense if Is(X, g)+ is adjoint hence embeds in GL(Lie(Is(X, g)+ )). Further, the “plus” on G(R)+ is necessary:
if Is(X, g)+ = SO(p, q)+ , this is not G(R) for algebraic G.)
Any p ∈ X is an isolated fixed point of the associated symmetry
sp ∈ Aut(X) with s2p = idX , and so dsp is multiplication by (−1) on
Tp X. In fact, by a delicate argument involving sectional curvature
(cf. [Milne2005, sec.1]), for any z = a + bi with |z| = 1, there exists an unique isometry up (z) of (X, g) such that on Tp X, dup (z) is
multiplication by z (i.e. a + bJ). Since dup yields a homomorphism
U1 → GL(Tp X), the uniqueness means that
up : U1 → Is(X, g)+
is also a homomorphism. It algebraizes to the homomorphism
φp : U → G
of real algebraic groups.
To view X as a conjugacy class, recall that G(R)+ acts transitively.
For g ∈ G(R)+ sending p 7→ q, the uniqueness of uq means that
φq (z) = g ◦ φp (z) ◦ g −1 = (Ψg ◦ φp )(z).
We must show that φp satisfies the three constraints. In the decomposition
gC = kC ⊕ Tp1,0 X ⊕ Tp0,1 X,
d(Ψφp (z) ) has eigenvalue z on T 1,0 hence z̄ = z −1 on T 0,1 , since φp is
real and z ∈ U1 . Using the uniqueness once more, Ψk ◦ φp = φp for any
k ∈ Kp , and so Ψφp (z) acts by the identity on k = Lie(Kp ). Therefore
1, z, z −1 are the eigenvalues of Adφp . Finally, from the fact that X
has negative sectional curvatures we deduce that Ψsp is Cartan, which
together with noncompactness of X implies that sp projects to e in no
factor of G.
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
9
C. Cartan’s classification of irreducible Hermitian symmetric
domains. Let X be an irreducible HSD, G the corresponding simple R-algebraic group, and T ⊂ GC a maximal algebraic torus. The
restriction to T
+

/ GC
/ GL(gC )
T
Ad
of the adjoint representation breaks into 1-dimensional eigenspaces on
which T acts through characters:
!
M
gC = t ⊕
gα ,
α∈R
where
R+ q R− = R ⊂ Hom(T, Gm ) ∼
= Zn
are the roots (with R− = −R+ ). With this choice, one has uniquely
the
+
• simple
P roots: {α1 , . . . , αn } such that each α ∈ R is of the
form
mi αi where m
Pi ≥ 0; and the
• highest root: α̂ =
m̂i αi ∈ R+ such that m̂i ≥ mi for any
+
other α ∈ R .
The αi give the nodes on the Dynkin diagram of G, in which αi and αj
are connected if they pair nontrivially under a standard inner product,
the Killing form
B(X, Y ) := T r(adXadY ).
Example 1.8.
An
•
•
•
•
•
•
Dn
•
E6
•
•
•
•
•
•
•
Over C, our circle map φ defines a cocharacter
µ
Gm

/
U
/
φC
)
GC .
This has a unique conjugate factoring through T in such a way that,
under the pairing of characters and cocharacters given by
Gm → T → Gm
µ
α
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
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z 7−→ z hµ,αi ,
one has hµ, αi ≥ 0 for all α ∈ R+ . Since µ must act through the
eigenvalues z, 1, z −1 , we know
hµ, αi = 0 or 1 for all α ∈ R+
.
and 6=
0
for some α ∈ R+
By considering hµ, α̂i, we deduce from this that hµ, αi i = 1 for a unique
i, and that the corresponding αi is special: i.e., m̂i = 1. So we have a
1-to-1 correspondence
irreducible
special nodes on
←→
,
Hermitian symmetric domains
connected Dynkin diagrams
and hence a list of the number of distinct isomorphism classes of irreducible HSD’s corresponding to each simple complex Lie algebra:
An Bn Cn Dn E6 E7 E8 F4 G2
n
1
1
3
2 1 0 0 0
Example 1.9.
(a) An : X ∼
= SU (p, q)/S(Up ×Uq ) with p+q = n+1 (n possibilities).
(b) Bn : X ∼
= SO(n, 2)+ /SO(n) × SO(2).
(c) Cn : X ∼
= Sp2n (R)/U (n) ∼
= {Z ∈ Mn (C) | Z = t Z, Im(Z) > 0}.
By J. Carlson’s lectures [Carlson], in (b) X is a period domain for
H 2 of K3 surfaces; in (c), X is the Siegel upper half-space Hn , and
parametrizes weight/level 1 Hodge structures.
D. Hodge-theoretic interpretation. Let V be a Q-vector space.
Definition 1.10. A Hodge structure on V is a homomorphism
ϕ̃ : S → GL(V )
defined over R, such that the weight homomorphism
ϕ̃
wϕ̃ : Gm ,→ S → GL(V )
is defined over Q.2
Associated to ϕ̃ is
µϕ̃ : Gm → GL(V )
z 7→ ϕ̃C (z, 1).
2These
are precisely the Q-split mixed Hodge structures (no nontrivial extensions). More generally, mixed Hodge structures have weight filtration W• defined
over Q but with the canonical splitting of W• defined over C, so that the weight
homomorphism is only defined over C.
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
11
Remark 1.11. Recalling that S(C) ∼
= C∗ × C∗ , V p,q ⊂ VC is the
(
z p wq - eigenspace of ϕ̃C (z, w)
,
zpeigenspace of µ(z)
and wϕ̃ (r) = ϕ̃(r, r) acts on it by rp+q .
Fix a weight n, Hodge numbers {hp,q }p+q=n , and polarization Q :
V × V → Q. Let
• D be the period domain parametrizing Hodge structures of this
type, polarized by Q, on V ;
• t ∈ ⊕i V ⊗ki ⊗ V̌ ⊗`i be a finite sum of Hodge tensors;
• Dt+ ⊂ D be a connected component of the subset of HS for
which these tensors are Hodge: ti ∈ F n(ki −`i )/2 for each i; and
• Mt ⊂ GL(V ) be the smallest Q-algebraic subgroup with Mt (R) ⊃
ϕ̃(S(R)) for all ϕ̃ ∈ Dt+ . (This is reductive.)
Then given any ϕ̃ ∈ Dt+ , the orbit
D+ = Mt (R)+ .ϕ̃ ∼
= Mt (R)+ /Hϕ̃
t
under action by conjugation, is called a Mumford-Tate domain.
This is a connected component of the full Mumford-Tate domain
Dt := Mt (R).ϕ̃, which will become relevant later (cf. §4.B). In either
(+)
case, Mt is the Mumford-Tate group of Dt .
Exercise 1.12. Check that Ψµϕ̃ (−1) is a Cartan involution.
Now, consider the condition that the tautological family V → Dt+ be
a variation of Hodge structure, i.e. that Griffiths’s infinitesimal period
relation (IPR) I ⊂ Ω• (Dt+ ) be trivial. This is equivalent to the statement that the HS induced on Lie(Mt ) ⊂ End(V ) “at ϕ̃” (by Ad ◦ ϕ̃)
be of type (−1, 1) + (0, 0) + (1, −1), since terms in the Hodge decomposition of type (−2, 2) or worse would violate Griffiths transversality.
Another way of stating this is that
(1.1)
Ad ◦ µϕ̃ (z) has only the eigenvalues z, 1, z −1 ,
and so we have proved part (a) of
Proposition 1.13. (a) A Mumford-Tate domain with trivial IPR (and
Mt adjoint) admits the structure of a Hermitian symmetric domain
with G defined over Q , and
(b) conversely — that is, such Hermitian symmetric domains parametrize VHS.
Remark 1.14. (i) Condition (1.1) implies that Ψµϕ̃ (−1) gives a symmetry of Dt+ at ϕ̃, but not conversely: e.g., an example of a Hermitian
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
12
symmetric MT domain with nontrivial IPR is the period domain for
HS of weight 6 and type (1, 0, 1, h, 1, 0, 1).
(ii) This doesn’t contradict (b), because the same Hermitian symmetric domain can have different MT domain structures.
(iii) Strictly speaking, to get the HSD structure in (a) one must put
φ := µϕ̃ ◦ , which unlike µϕ̃ is actually defined over R.
(iv) The isotropy group Hϕ̃ above is a maximal compact subgroup
of Mt (R)+ iff Dt+ is Hermitian symmetric.
φ
Proof. To show (b), let X be a HSD with real circle U → G. Since a
product of MT domains is a MT domain, we may assume G Q-simple.
The composition
(z, w)
S
/
/ z/w
U
/
φ
G
/
Aut(g, B)
8
=:ϕ̃
is a Hodge structure on V = g, polarized by −B since Ψφ(−1) is Cartan.
The Q-closure of a generic G(R)-conjugate is G by nontriviality of
φ(−1), and G is of the form Mt by Chevalley’s theorem (cf. Griffiths’s
lectures in this volume). Consequently, X = G(R)+ .ϕ̃ is a MT domain.
The IPR vanishes because Ad ◦ φ has eigenvalues z, 1, z −1 .
The proposition is essentially a theorem of [Deligne1979].
Example 1.15. [due to Mark Green] Applied to one of the HSD’s for
E6 , this procedure yields a MT domain parametrizing certain HS of
type (h2,0 , h1,1 , h0,2 ) = (16, 46, 16) — a submanifold Dt+ of the period
domain D for such HS. The IPR I ⊂ Ω• (D) is nontrivial but pulls back
to zero on Dt+ .
Problem 1.16. Find a family of varieties over Dt+ with this family of
HS. More generally, it is conjectured that the tautological VHS over every MT domain with trivial IPR, is motivic – i.e., comes from algebraic
geometry.
The proof of (b) always produces HS of even weight. Sometimes, by
replacing the adjoint representation by a “standard” representation, we
can parametrize HS of odd weight: For instance, H4 ∼
= Sp8 (R)/U (4)
parametrizes HS of weight/level 1 and rank 8, or equivalently abelian
varieties of dimension 4. There are “two” types of MT subdomains in
H4 :
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
13
(a) Those corresponding to EndHS (V ) (= End(A)Q ) containing a nontrivial fixed subalgebra E isomorphic to a product of matrix algebras
over Q-division algebras. These must be of the four types occurring in
the Albert classification:
(I) totally real field;
(II) indefinite quaternion algebra over a totally real field;
(III) definite quaternion algebra over a totally real field;
(IV) division algebra over a CM field.
For example, an imaginary quadratic field is a type (IV) division algebra. All four types do occur in H4 .
(b) Those corresponding to fixed endomorphisms E plus higher Hodge
tensors. We regard EndHS (V ) as a subspace of T 1,1 V and the polarization Q as an element of T 0,2 V ; “higher” means [in a T k,` V ] of degree
k + ` > 2.
Here are two such examples:
Example 1.17. (Type (a)) Fix an embedding
β
Q(i) ,→ End(V ),
and write Hom(Q(i), C) = {η, η̄}. We consider Hodge structures on V
such that
V 1,0 = Vη1,0 ⊕ Vη̄1,0
with dim Vη1,0 = 1; in particular, for such HS the image of Q(i) lies
in EndHS (V ). The resulting MT domain is easily presented in the
three forms from §1.B: as the circle conjugacy class Dt+ (taking t :=
{β(i), Q}); as the noncompact Hermitian space SU (1, 3)/S(U1 × U3 );
and as a complex 3-ball. It is irreducible of Cartan type A3 .
Exercise 1.18. Show that the MT group of a generic HS in Dt+ has
real points M (R)/wϕ̃ (R∗ ) ∼
= U (2, 1).
Example 1.19. (Type (b)) [Mumford1969] constructs a quaternion
algebra Q over a totally real cubic field K, such that Q ⊗Q R ∼
= H⊕
H ⊕ M2 (R), together with an embedding Q∗ ,→ GL8 (Q). This yields a
Q-simple algebraic group
G := ResK/Q UQ ⊂ Sp8 ⊂ GL(V )
with G(R) ∼
= SU (2)×2 × SL2 (R), and the G(R)-orbit of
ϕ0 : U → G
a b
×2
a + ib 7→ id ×
−b a
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
14
yields a MT domain. Mumford shows that the generic HS it supports
has trivial endomorphisms E = Q, so G is cut out by higher Hodge
tensors.
Remark 1.20. Though we limited ourselves to pure HS above to simplify
the discussion, the Mumford-Tate business, and Proposition 1.13(a) in
particular, still works in the more general setting of Definition 1.10.
2. Locally symmetric varieties
To construct quotients of Hermitian symmetric domains we’ll need
the basic
Proposition 2.1. Let X be a topological space, with x0 ∈ X; G be a
locally compact group acting on X; and Γ ≤ G be a discrete subgroup
(i.e. one with no limit points). Assume
(i) K := stab(x0 ) is compact, and
(ii) gK 7→ gx0 : G/K → X is a homeomorphism.
Then Γ\X is Hausdorff.
The proof is a nontrivial topology exercise. Writing π : X → Γ\X,
the key points are:
• π −1 of a compact set is compact; and
• the intersection of a discrete and compact set is finite.
Corollary 2.2. Let X = G(R)+ /K be a Hermitian symmetric domain,
and Γ ≤ G(R)+ discrete and torsion-free. Then Γ\X has a unique
complex-manifold structure for which π is a local isomorphism.
Remark 2.3. If Γ isn’t torsion-free then we get an orbifold.
Example 2.4. (a) X = H, G = SL2 acting in the standard way, and
Γ = Γ(N ) := ker{SL2 (Z) → SL2 (Z/N Z)}
with N ≥ 3. The quotients Γ\X =: Y (N ) are the classical modular
curves classifying elliptic curves with marked N -torsion, an example
of level structure.
(b) X = Hn , G = Sp2n , Γ = Sp2n (Z). Then Γ\X is the Siegel
modular variety classifying abelian n-folds with a fixed polarization.
(c) X = H × · · · × H (n times), G = ResF/Q SL2 with F a totally
real field of degree n over Q, and Γ = SL2 (OF ). The quotient Γ\X is
a Hilbert modular variety classifying abelian n-folds with E ⊃ F :
the general member is of Albert type (I). We may view X as a proper
MT subdomain of Hn . To get a more interesting level structure here,
one could replace OF by a proper ideal.
SHIMURA VARIETIES:
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15
P
(d) X = {[v] ∈ Pn (C) | −|v0 |2 + ni=1 |vi |2 < 0} (the complex n-ball),
G = ResK/Q SU (n, 1) with K a quadratic imaginary field (Hom(K, C) =
{θ, θ̄}) and Γ = SU ((n, 1), OK ). The quotient Γ\X is a Picard modular variety, classifying abelian (n + 1)-folds with E ⊃ K in such a
way that the dimension of the θ- resp. θ̄-eigenspaces in T0 A are 1 resp.
n. See the interesting treatment for n = 2 in [Holzapfel1995], especially
sec. 4.9.
All the discrete groups Γ arising here have been rather special.
Definition 2.5. (a) Let G be a Q-algebraic group. Fix an embedding
ι
G ,→ GLn . A subgroup Γ ≤ G(Q) is
arithmetic ⇐⇒ Γ commensurable3 with G(Q) ∩ GLn (Z); and
congruence ⇐⇒
for some N, Γ contains
Γ(N ) := G(Q) ∩ {g ∈ GLn (Z) | g ≡ id.}
(N )
as a subgroup of finite index.
Congruence subgroups are arithmetic, and both notions are independent of the embedding ι.
(b) Let G be a connected real Lie group. A subgroup Γ ≤ G is
arithmetic if there exist
• a Q-algebraic group G,
• an arithmetic Γ0 ≤ G(Q), and
• a homomorphism G(R)+ G with compact kernel,
ρ
such that ρ(Γ0 ∩ G(R)+ ) = Γ.
Part (b) is set up so that Γ will always contain a torsion-free subgroup
of finite index.
Theorem 2.6. [BB1966] Let X = G(R)+ /K be a Hermitian symmetric
domain, and Γ < G(R)+ a torsion-free arithmetic subgroup. Then
X(Γ) := Γ\X is canonically a smooth quasi-projective algebraic variety,
called a locally symmetric variety.
Remark 2.7. If we don’t assume Γ torsion-free, we still get a quasiprojective algebraic variety, but it is an orbifold, hence not smooth
and not called a locally symmetric variety.
Idea of proof. Construct a minimal, highly singular (Baily-Borel) compactification
X(Γ)∗ := Γ \{X q B}
3i.e.
the intersection is of finite index in each
SHIMURA VARIETIES:
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16
where B stands for “rational boundary components”. Embed this in PN ,
using automorphic forms of sufficiently high weight, as a projective
analytic — hence (by Chow/GAGA) projective algebraic — variety.
The existence of enough automorphic forms to yield an embedding is
a convergence question for certain Poincaré-Eisenstein series.
Example 2.8. In the modular curve context (with Γ = Γ(N )), B =
P1 (Q) and X(Γ)∗ \X(Γ) is a finite set of points called cusps. We write
Y (N ) resp. X(N ) for X(Γ) resp. X(Γ)∗ .
Recalling from §1.D that X is always a MT domain, we can give a
Hodge-theoretic interpretation to the Baily-Borel compactification:
Proposition 2.9. The boundary components B parametrize the possible ⊕i GriW Hlim for VHS into X(Γ).
Heuristic idea of proof. Assuming P GL2 is not a quotient of G, the au⊗N
tomorphic forms are Γ-invariant sections of KX
for some N 0. The
V
canonical bundle KX , which is pointwise isomorphic to d g(−1,1) , measures the change of the Hodge flag in every direction. So the boundary
components parametrized by these sections must consist of naive limiting Hodge flags in ∂ X̄ ⊂ X̌. In that limit, thinking projectively, the
relation between periods that blow up at different rates (arising from
different GriW ) is fixed, which means we cannot see extension data. On
the other hand, since exp(zN ) does not change the GriW F • , this information is the same for the naive limiting Hodge flag and the limiting
mixed Hodge structure.
Remark 2.10. (a) The proposition is due to Carlson, Cattani and Kaplan [CCK1980] for Siegel domains, but there seems to be no reference
for the general statement.
(b) There are other compactifications with different Hodge-theoretic
interpretations:
• the compactification of Borel-Serre [BS1973], which records GriW
and adjacent extensions, at least in the Siegel case; and
• the smooth toroidal compactifications of [AMRT1975], which
capture the entire limit MHS.
The latter is what the monograph of Kato and Usui [KU2009] generalizes (in a sense) to the non-Hermitian symmetric case, where X(Γ) is
not algebraic.
For the discussion of canonical models to come, we will need the
Theorem 2.11. [Borel1972] Let Y be a quasi-projective algebraic variety over C, and X(Γ) a locally symmetric variety. Then any analytic
map Y → X(Γ) is algebraic.
SHIMURA VARIETIES:
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17
Idea of proof. Extend this to an analytic map Ȳ → X(Γ)∗ , then use
GAGA.
Suppose X = H and Y is a curve; since Γ is torsion-free, X(Γ) ∼
=
C\{≥ 2 points}. Denote by D a small disk about the origin in C. If
a holomorphic f : (D\{0}) → X(Γ) does not extend to a holomorphic
map from D to P1 , then f has an essential singularity at 0. By the
“big” Picard theorem, f takes all values of C except possibly one, a
contradiction. Applying this argument to a neighborhood of each point
of Ȳ\Y gives the desired extension.
The general proof uses the existence of a good compactification Y ⊂
Ȳ (Hironaka) so that Y is locally D×k × (D∗ )×` .
3. Complex multiplication
A. CM Abelian varieties. A CM field is a totally imaginary field
E possessing an involution ρ ∈ Gal(E/Q) =: GE such that φ ◦ ρ = φ̄
for each φ ∈ Hom(E, C) =: HE .
Exercise 3.1. Show that then E ρ is totally real, and ρ ∈ Z(GE ).
Denote by E c a normal closure.
For any decomposition
HE = Φ q Φ̄,
(E, Φ) is a CM type; this is equipped with a reflex field
o
n P
φ(e)
e
∈
E
⊂ Ec
E 0 := Q
φ∈Φ
n
o
c
= fixed field of σ ∈ GE | σ Φ̃ = Φ̃
where Φ̃ ⊂ HE c consists of embeddings restricting on E to those in
Φ. Composing Galois elements with a fixed choice of φ1 ∈ Φ gives an
identification
∼
=
HE c ←− GE c
and a notion of inverse on HE c . Define the reflex type by
n
o
0
−1
Φ := φ̃ |E 0 φ̃ ∈ Φ̃ ,
and reflex norm by
NΦ0 : (E 0 )∗ → E ∗
Y
e0 7→
φ0 (e0 ).
φ0 ∈Φ0
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
18
√
Example 3.2. (a) All imaginary quadratic fields Q( −d) are CM; in
this case NΦ0 is the identity or complex conjugation.
(b) All cyclotomic fields Q(ζn ) are CM; and if E/Q is an abelian
extension, then E 0 = E c = E and E is contained in some Q(ζn ). For
cyclotomic fields we will write
φj := embedding sending ζn 7→ e2πij/n .
(c) The CM type (Q(ζ5 ); {φ1 , φ2 }) has reflex (Q(ζ5 ); {φ1 , φ3 }).
The relationship of this to algebraic geometry is contained in
Proposition 3.3. (a) For a simple complex abelian g-fold A, the following are equivalent:
(i) the MT group of H 1 (A) is a torus;
(ii) End(A)Q has (maximal) rank 2g over Q;
(iii) End(A)Q is a CM field; and
(E,Φ)
for some CM type (E, Φ) and ideal
(iv) A ∼
= Cg /Φ(a) =: Aa
a ⊂ OE .
(E,Φ)
(b) Furthermore, any complex torus of the form Aa
is algebraic.
A CM abelian variety is just a product of simple abelian varieties,
each satisfying the conditions in (a). We will suppress the superscript
(E, Φ) when the CM type is understood. Note that in (i), the torus
may be of dimension less than g, the so-called degenerate case.
Example 3.4. Φ(a) means the 2g-lattice



 φ1 (a) 
 ...  a ∈ a .


φg (a) For Example 3.2 (c) above,
2πi/5 4πi/5 6πi/5 1
e
e
e
2
AOQ(ζ5 ) = C
Z
,
,
,
.
4πi/5
3πi/5
1
e2πi/5
e
e
The interesting points in Proposition 3.3 are:
• where does the CM field come from?
(E,Φ)
polarized?
• why is Aa
In lieu of a complete proof we address these issues.
Proof of (i) =⇒ (iii). V = H 1 (A) is polarized by some Q. Set
E := EndHS (H 1 (A)) = ZGL(V ) (M ) (Q) ∪ {0},
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
19
where the centralizer ZGL(V ) (M ) is a Q-algebraic group and contains
a maximal torus T . Since T commutes with M and is maximal, T
contains M (possibly properly). One deduces that
• T (Q) ∪ {0} =: E (,→ E) is a field,
η
• V is a 1-dimensional vector space over E, and
• E is actually all of E.
M diagonalizes with respect to a Hodge basis
ω1 , . . . , ωg ; ω̄1 , . . . , ω̄g
√
such that −1Q(ωi , ω̄j ) = δij . The maximal torus in GL(V ) this basis
defines, centralizes M hence must be T .
Now write HE = {φ1 , . . . , φ2g }, E = Q(ξ) and
mξ (λ) =
2g
Y
(λ − φi (ξ))
i=1
for the minimal polynomial of ξ, hence η(ξ). Up to reordering, we
therefore have
[η(ξ)]ω = diag({φi (ξ)}2g
i=1 )
for the matrix of “multiplication by ξ” with respect to the Hodge basis.
Since η(ξ) ∈ GL(V ) (a fortiori ∈ GL(VR )), and φj (ξ) determines φj ,
ωi+g = ω̄i
=⇒
φi+g = φ̄i .
Define the Rosati involution † : E → E by
Q(ε† v, w) = Q(v, εw)
∀v, w ∈ V.
This produces ρ := η −1 ◦ † ◦ η ∈ GE , and we compute
φi+g (e)Q(ωi , ωi+g ) = Q(ωi , η(e)ωi+g ) = Q(η(e)† ωi , ωi+g )
= Q(η(ρ(e))ωi , ωi+g ) = φi (ρ(e))Q(ωi , ωi+g ),
which yields φi ◦ ρ = φ̄i .
Proof of (b). We have the following construction of H 1 (A): let V be a
2g-dimensional Q-vector space with identification
∼
=
β: E→V
inducing (via multiplication in E)
η : E ,→ End(V ).
Moreover, there is a basis ω = {ω1 , . . . , ωg ; ω̄1 , . . . , ω̄g } of VC with respect to which
[ηC (e)]ω = diag{φ1 (e), . . . , φg (e); φ̄1 (e), . . . , φ̄g (e)},
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
20
and we set V 1,0 := C hω1 , . . . , ωg i. This gives η(e) ∈ EndHS (V ) and
V ∼
= H 1 (A).
HS
√
Now, there exists a ξ ∈ E such that −1φi (ξ) > 0 for i = 1, . . . , g,
and we can put
Q(β(e), β(ẽ)) := T rE/Q (ξ · e · ρ(ẽ)) : V × V −→ Q.
Over C, this becomes

φ1 (ξ)
0


[Q]ω = 


..
.
φg (ξ)
φ̄1 (ξ)
..
0
.


.

φ̄g (ξ)
Remark 3.5. NΦ0 algebraizes to a homomorphism of algebraic groups
NΦ0 : ResE 0 /Q Gm → ResE/Q Gm
which gives NΦ0 on the Q-points, and the MT group
MH 1 (A) ∼
= image(NΦ0 ).
Let E be a CM field with [E : Q] = 2g. In algebraic number theory
we have
I(E)
the monoid of nonzero ideals in OE ,
J (E)
the group of fractional ideals
(of the form e · I, e ∈ E ∗ and I ∈ I(E)), and
P(E)
the subgroup of principal fractional ideals
(of the form (e) := e · OE , e ∈ E ∗ ).
The (abelian!) ideal class group
Cl(E) :=
J (E)
,
P(E)
or more precisely the class number
hE := |Cl(E)|,
expresses (if 6= 1) the failure of OE to be a principal ideal domain (and
to have unique factorization). Each class τ ∈ Cl(E) has a representative I ∈ I(E) with norm bounded by the Minkowski bound, which
implies hE is finite.
Now let
n
o
(E,Φ) Aa
a ∈ J (E)
Ab(OE , Φ) :=
,
isomorphism
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
21
where the numerator denotes abelian g-folds with OE ⊂ End(A) acting
on T0 A through Φ. They are all isogenous, and multiplication by any
e ∈ E ∗ gives an isomorphism
Aea .
Aa −→
∼
=
Hence we get a bijection
Cl(E) −→
Ab(OE , Φ)
∼
.
Aa
√
Example 3.6. The class number of Q( −5)
Representatives
is 2. √
of√ √
Ab(OQ( −5) , {φ}) are the elliptic curves C/Z 2, 1 + −5 and C/Z 1, −5 .
=
[a]
7−→
The key point now is to notice that for A ∈ Ab(OE , Φ) and σ ∈
Aut(C),
σ
A ∈ Ab(OE , σΦ).
This is because endomorphisms are given by algebraic cycles, so that
the internal ring structure of E is left unchanged by Galois conjugation; what changes are the eigenvalues of its action on T0 A. From the
definition of E 0 we see that
Gal(C/E 0 ) acts on Ab(OE , Φ),
which suggests that the individual abelian varieties should be defined
over an extension of E 0 of degree hE 0 . This isn’t exactly true if Aut(A) 6=
{id}, but the argument does establish that any CM abelian variety is
defined over Q̄.
B. Class field theory. In fact, it gets much better: not only is there
a distinguished field extension HL /L of degree hL for any number field
L; there is an isomorphism
(3.1)
Gal(HL /L) o
∼
=
Cl(L) .
To quote Chevalley, “L contains within itself the elements of its own
transcendence”.
Idea of the construction of (3.1). Let L̃/L be an extension of degree d
which is
• abelian: Galois with Gal(L̃/L) abelian
Q
• unramified: for each prime p ∈ I(L), p.OL̃ = ri=1 Pi for some
r|d.
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
22
Writing N (p) := |OL /p|, the extension (OL̃ /Pi )/(OL /p) of finite fields
has degree N (p)d/r . The image of its Galois group in Gal(L̃/L) by
n
o
(OL̃ /Pi ) o
/ Gal(L̃/L)
σ ∈ Gal(L̃/L) σPi = Pi Gal( (OL /p) )
∼
O
=
generated
by
α → αN (p)
(mod Pi )
/
=: F robp
is (as the notation suggests) independent of i, yielding a map from
prime ideals
−→ Gal(L̃/L).
of L
Taking L̃ to be the Hilbert class field
HL := maximal unramified abelian extension of L,
this leads (eventually) to (3.1).
More generally, given I ∈ I(L), we have the ray class group mod
I
Cl(I) =
fractional ideals prime to I
principal fractional ideals with generator ≡ 1 (mod I)
and ray class field mod I
LI = the maximal abelian extension of L in which
all primes ≡ 1 mod I split completely.
(Morally, LI should be the maximal abelian extension in which primes
dividing I are allowed to ramify, but this isn’t quite correct.) There is
an isomorphism
Gal(LI /L) ←−
Cl(I),
∼
=
and LI ⊇ HL with equality when I = OL .
Example 3.7. For L = Q, L(n) = Q(ζn ).
To deal with the infinite extension
Lab := maximal abelian extension of L (⊂ Q̄),
of L, we have to introduce the adéles.
In studying abelian varieties one considers, for ` ∈ Z prime, the finite
groups of `-torsion points A[`]; multiplication by ` gives maps
· · · → A[`n+1 ] → A[`n ] → · · · .
SHIMURA VARIETIES:
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23
If we do the same thing on the unit circle S 1 ⊂ C∗ , we get
···
→ S 1 [`n+1 ]
k
→ Z/`n+1 Z
·`
S 1 [`n ] → · · ·
k
Z/`n Z →
→
→
natural
map
and one can define the `-adic integers by the inverse limit
Z` := lim Z/`n Z.
←−
n
An element of this limit is by definition an infinite sequence of elements
in the Z/`n Z mapping to each other. There is the natural inclusion
Z ,→ Z` ,
and
Q` := Z` ⊗Z Q.
P
Elements of Q` can be written as power series i≥n ai `i for some n ∈ Z
(n ≥ 0 for elements of Z` ). Q` can also be thought of as the completion
of Q with respect to the metric given by
1
a
if x − y = `n with a, b relatively prime to `.
n
`
b
The resulting topology on Z` makes
d(x, y) =
Un (α) := {α + λ`n | λ ∈ Z` }
into “the open disk about α ∈ Z` of radius `1n ”. Z` itself is compact and
totally disconnected.
Now set
Ẑ := lim Z/nZ,
←−
n∈Z
Q
which is isomorphic to ` Z` by the Chinese remainder theorem. The
finite adéles appear naturally as
Y0
Af := Ẑ ⊗Z Q =
Q` ,
`
Q0
where
means the ∞-tuples with all but finitely many entries in Z` .
The “full” adéles are constructed by writing
AZ := R × Ẑ
AQ = Q ⊗Z AZ = R × Af ,
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
24
which generalizes for a number field L to
Y
AL = L ⊗Q AQ = R[L:Q] ×
0
LP ,
P∈I(L)
|
{z
AL,f
}
Q
where 0 means all but finitely many entries in (OL )P .
For a Q-algebraic group G, we can define
Y0
G(Af ) :=
G(Q` )
G(AQ ) := G(R) × G(Af )
Q
with generalizations to AL and AL,f . Here 0 simply means that for
some (hence every) embedding G ,→ GLN , all but finitely many entries
lie in G(Z` ). The idéles
A×
(f ) = Gm (A(f ) )
A×
L(, f ) = (ResL/Q Gm )(A(f ) ) = Gm (AL(, f ) )
were historically defined first; Weil introduced “adéle” – also (intentionally) a girl’s name – as a contraction of “additive idéle”. The usual
norm NL̃/L and reflex norm NΦ0 extend to maps of idéles, using the
formulation of these maps as morphisms of Q-algebraic groups.
Returning to S 1 ⊂ C∗ , let ζ be an N th root of unity and a = (an ) ∈ Ẑ;
then
ζ 7→ ζ a := ζ aN defines an action of Ẑ
on the torsion points of S 1 (which generate Qab ).
The cyclotomic character
∼
=
χ : Gal(Qab /Q) −→ Ẑ× ∼
= Q× \A×
Q,f
is defined by
σ(ζ) =: ζ χ(σ) ,
and we can think of it as providing a “continuous envelope” 4 for the action of a given σ on any finite order of torsion. The Artin reciprocity
map is simply its inverse
artQ := χ−1
4I
use this term because the automorphisms of C other than complex conjugation
induce highly discontinuous (non-measurable!) maps on the complex points of a
variety over Q. But if one specifies a discrete set of points, it is sometimes possible
to produce a continuous (even analytic/algebraic) automorphism acting in the same
way on those points.
SHIMURA VARIETIES:
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25
for Q. Assuming now that L is totally imaginary, this generalizes to
artL
L× \A×
L,f
×
L
A×
L,f
//
Gal(Lab /L)
(#)
NL̃/L (A×
)
L̃,f
∼
=
/
Gal(L̃/L).
If L̃ = LI then the double-coset turns out to be Cl(I), so this recovers
the earlier maps for ray class fields. (For L̃ = HL , we can replace
cL .) The correspondence between
NL̃/L (A×
) by O
L̃,f
open subgroups
finite abelian
←→
of A×
extensions
of L
L,f
is the essence of class field theory. Also note that (#) gives compatible
maps to all the class groups of L, so that A×
L,f acts on them.
C. Main Theorem of CM. Now let’s bring the adéles to bear upon
abelian varieties. Taking the product of the Tate modules
n
T` A := ←
lim
−A[` ]( = rank-2g free Z` -module)
n
of an abelian g-fold yields
Tf A :=
Y
T` A,
`
Vf A := Tf A⊗Z Q( = rank-2g free Af -module)
with (for example)
Aut(Vf A) ∼
= GL2g (Af ).
The “main theorem”, due to Shimura and Taniyama, is basically a
detailed description of the action of Gal(C/E 0 ) on Ab(OE , Φ) and the
torsion points of the (finitely many isomorphism classes of) abelian
varieties it classifies.
Theorem 3.8. Let A[a] ∈ Ab(OE , Φ) and σ ∈ Gal(C/E 0 ) be given. For
any a ∈ A×
E 0 ,f with artE 0 (a) = σ|(E 0 )ab , we have:
σ
(a) A[a] ∼
= ANΦ0 (a).[a] (where NΦ0 (a) ∈ A×
E,f , and NΦ0 (a).[a] depends
only on σ|HE0 ); and
(b) there exists a unique E-linear isogeny α : A[a] → σ A[a] such that
α (NΦ0 (a).x) = σ x
(∀x ∈ Vf A).
SHIMURA VARIETIES:
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26
Sketch of proof for (b).
• σ A ∈ Ab(OE , Φ) =⇒ ∃ (E-linear) isogeny α : A → σ A;
• Vf A is free of rank 1 over AE,f ;
Vf (α)−1
σ
• the composition Vf (A) −→ Vf (σ A) −→ Vf (A) is AE,f -linear, so is
just multiplication by some s ∈ A×
E,f ;
• s is independent (up to E × ) of the choice of α, defining the horizontal
arrow of
(3.2)
/
Gal(C/E 0 )
E × \A×
E,f
O
e
((
Gal((E 0 )ab /E 0 )
OO
(∗)
artE 0
(E 0 )× \A×
E 0 ,f
• A is defined over a number field k; the Shimura-Taniyama computation of the prime decompositions of the elements of E ∼
= End(A)Q
reducing to various Frobenius maps (in residue fields of k) then shows
that the vertical map (∗) in (3.2) is NΦ0 . Hence
NΦ0 (a) = s = Vf (α)−1 ◦ σ
which gives the formula in (b).
So what does (b) mean? Like the cyclotomic character, we get a very
nice interpretation when we restrict to the action on m-torsion points
of A for any fixed m ∈ N:
Corollary 3.9. There exists a unique E-linear isogeny αm : A → σ A
such that
αm (x) = σ x
(∀x ∈ A[m]).
That is, αm provides a “continuous envelope” for the action of automorphisms of C on special points.
4. Shimura varieties
A. Three key adelic lemmas. Besides the main theorem of CM,
there is another (related) connection between the class field theory
described in III.B and abelian varieties. The tower of ray class groups
associated to the ideals of a CM field E can be expressed as
(4.1)
E × A× UI ∼
= T (Q) \T (Af )/ UI
E,f
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
27
where
(
UI :=
(ap )p∈I(E) prime ∈ A×
E,f
ap ∈ (OE )p
ap ≡ 1 mod pordp I
for all p,
)
for the finitely
many p dividing I
is a compact open subgroup of A×
E,f = T (Af ) and
T = ResE/Q Gm .
(4.1) may be seen as parametrizing abelian varieties with CM by a
type (E, Φ) and having fixed level structure — which “refines” the set
parametrized by Cl(E). Shimura varieties give a way of extending
this story to more general abelian varieties with other endomorphism
and Hodge-tensor structures, as well as the other families of Hodge
structures parametrized by Hermitian symmetric domains.
The first fundamental result we will need is
Lemma 4.1. For T any Q-algebraic torus, and Kf ⊂ T (Af ) any open
subgroup, T (Q)\T (Af )/Kf is finite.
Sketch of Proof. This follows from the definition of compactness if we
can show T (Q)\T (Af ) is compact. For any number field F , the latter is
closed in T (F )\T (AF,f ), and for some F over which T splits the latter
dim(T )
is (F × \A×
. Finally, by the Minkowski bound
F,f )
c
F × \A×
F,f /OF = Cl(F ) is finite,
cF is compact (like Ẑ), so F × \A× is compact.
and O
F,f
For a very different class of Q-algebraic groups, we have the contrasting
Lemma 4.2. Suppose G/Q is semisimple and simply connected, of
noncompact type5; then
(a) [Strong approximation] G(Q) ⊆ G(Af ) is dense, and
(b) For any open Kf ⊆ G(Af ), G(Af ) = G(Q) · Kf .
Remark 4.3. Lemma 4.2(b) implies that the double coset G(Q)\G(Af )/Kf
is trivial. Note that double cosets are now essential as we are in the
nonabelian setting.
Sketch of (a) =⇒ (b). Given (γ` ) ∈ G(Af ), U := (γ` ) · Kf is an open
subset of G(Af ), hence by (a) there exists a g ∈ U ∩ G(Q). Clearly
g = (γ` ) · k for some k ∈ Kf , and so (γ` ) = g · k −1 .
5i.e.
none of its simple almost-direct Q-factors Gi have Gi (R) compact
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
28
Nonexample 4.4. Gm , which is of course reductive but not semisimple. If (b) held, then the ray class groups of Q (which are ∼
= (Z/`Z)× )
would be trivial. But even more directly, were Q× dense in A×
f , there
Q ×
×
would be q ∈ Q Q
close to any (a` ) ∈ Z` . This forces [the image in
A×
of]
q
to
lie
in
Z×
f
` , which means for each ` that (in lowest terms)
the numerator and denominator of q are prime to `. So q = ±1, a
contradiction.
Finally, for a general Q-algebraic group G, we have
Lemma 4.5. The congruence subgroups of G(Q) are precisely the Kf ∩
G(Q) (intersection in G(Af )) for compact open Kf ⊆ G(Af ).
Sketch of Proof. For N ∈ N the
K(N ) := (g` )` prime ∈ G(Af )
g` ∈ G(Z` )
for all `,
g` ≡ e mod `ord` N for each `|N
are compact open in G(Af ), and
g` ≡ e mod `ord` N
g ∈ G(Z) K(N ) ∩ G(Q) =
for each `|N g ∈ G(Z) g ≡ e mod N
=
= Γ(N ).
In fact, the K(N ) are a basis of open subsets containing e. So any
compact open Kf contains some K(N ), and
(Kf ∩ G(Q))/ (K(N ) ∩ G(Q)) ⊆ Kf /K(N )
is a discrete subgroup of a compact set, and therefore finite.
In some sense, Kf is itself the congruence condition.
Shimura datum (SD)
X̃
B. Shimura data. A
is a pair G,
[resp.] connected SD
X
consisting of
reductive
• G :=
algebraic group defined over Q
semisimple
and
G(R)X̃
•
:= a
conjugacy class of homomorphisms
Gad (R)+ X
GR
ϕ̃ : S →
,
Gad
R
satisfying the axioms:
(SV1)
only z/z̄, 1, z̄/z occur as eigenvalues of
Ad ◦ ϕ̃ : S → GL Lie(Gad )C ;
SHIMURA VARIETIES:
(SV2)
(SV3)
A HODGE-THEORETIC PERSPECTIVE
29
ΨAd(ϕ̃(i)) ∈ Aut(Gad
R ) is Cartan;
ad
G has no Q-factoron which the projection
X̃
of every (Ad ◦) ϕ̃ ∈
is trivial
X
wϕ̃
(SV4)
(SV5)
the weight homomorphism Gm

/S
ϕ̃
/' G
(a priori defined over R) is defined over Q;
(Z ◦ /wϕ̃ (Gm )) (R) is compact;6
and
(SV6)
Z ◦ splits over a CM field.
In the “connected” case, SV4-6 are trivial, while SV1-3 already imply:
• X is a Hermitian symmetric domain (in the precise sense of III
in §1.B, with φ = µϕ̃ ◦ ); and
• G is of noncompact type [from SV3], but with ker (G(R)+ Hol(X)+ )
compact [from SV2],
where the surjective arrow is defined by the action on the conjugacy
class of ϕ̃.
In these definitions, axioms SV4-6 are sometimes omitted: for example, canonical models exist for Shimura varieties without them. We
include them here from the beginning because they hold in the context
of Hodge theory. Indeed, consider a full Mumford-Tate domain X̃ (cf.
§1.D) for polarized Q-Hodge structures with generic Mumford-Tate
group G. Regarded as a pair, (G, X̃) always satisfies
SV2:
SV3:
SV4:
SV5:
SV6:
6SV5
by the second Hodge-Riemann bilinear relation (cf. Exercise 1.12);
otherwise the generic MT group would be a proper subgroup of G;
because the weight filtration is split over Q;
as G is a Mumford-Tate group, G/wϕ̃ (Gm ) contains a G(R)+ conjugacy class of anisotropic maximal real tori, and these
contain ZR◦ /wϕ̃ (Gm );
since G is defined over Q, the conjugacy class contains tori
defined over Q; and so X̃ contains a ϕ̃ factoring through
some such rational T . This defines a polarized CM Hodge
structure, with MT group a Q-torus T0 ⊆ T split over a
CM field (cf. §3). If T0 + Z ◦ , then the projection of ϕ̃ to
some Q -factor of Z ◦ is trivial and then it is trivial for all
is sometimes weakened: cf. [Milne2005, sec. 5].
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
30
its conjugates, contradicting SV3. Hence T0 ⊇ Z ◦ and Z ◦
splits over the CM field.
Further, SV1 holds if and only if the IPR on X̃ is trivial.
What can one say about an arbitrary Shimura datum? First, any
SD produces a connected SD by
• replacing G by Gder (which has the same Gad )
• replacing X̃ by a connected component X ( which we may view
as a Gad (R)+ -conjugacy class of homomorphisms Ad ◦ ϕ̃),
and so
X̃ is a finite union of Hermitian symmetric domains.
Moreover, by SV2 and SV5 one has that Ψϕ̃(i) ∈ Aut(GR /wϕ̃ (Gm )) is
Cartan, so that there exist a symmetric bilinear form Q on a Q-vector
space V , and an embedding ρ : G/wϕ̃ (Gm ) ,→ Aut(V, Q), such that
Q(·, (ρ◦ϕ̃)(i)·) > 0. As in the “Hodge domains” described in the lectures
by Griffiths, for any such faithful representation ρ̃ : G ,→ GO(V, Q)
with Q polarizing ρ̃ ◦ ϕ̃, X̃ is realized as a finite union of MT domains
with trivial IPR. Let M denote the MT group of X̃. If G = M , then
X̃ is exactly a “full MT domain” as described in §1.D. However, in
general M is a normal subgroup of G with the same adjoint group;
that the center can be smaller is seen by considering degenerate CM
Hodge structures. Suffice it to say that (while the correspondence is
slightly messy) all Shimura data, hence ultimately all Shimura varieties
as defined below, have a Hodge-theoretic interpretation.
Now given a connected SD (G, X), we add one more ingredient: let
Γ ≤ Gad (Q)+
be a torsion-free arithmetic subgroup, with inverse image in G(Q)+ a
congruence subgroup. Its image Γ̄ in Hol(X)+ is
(i)
(ii)
[torsion-free and] arithmetic: since ker(G(R)+ Hol(X)+ )
compact
isomorphic to Γ: Γ ∩ ker(G(R)+ Hol(X)+ ) =discrete ∩
compact = finite, hence torsion (and there is no torsion).
We may write
X(Γ) := Γ\X = Γ̄\X ;
(ii)
and Γ̄\X is a locally symmetric variety by (i) and Baily-Borel. By
Borel’s theorem,
(4.2)
Γ ≤ Γ0 =⇒ X(Γ0 ) X(Γ) is algebraic.
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
31
Definition 4.6. The connected Shimura variety associated with
(G, X, Γ) is
Sh◦Γ (G, X) := X(Γ).
Remark 4.7. Every X(Γ) is covered by an X(Γ0 ) with Γ0 the image of a
congruence subgroup of G(Q)+ . If one works with “sufficiently small”
congruence subgroups of G(Q), then
• they belong to G(Q)+ ;
• they have torsion-free image in Gad (Q)+ ;
• congruence =⇒ arithmetic.
This will be tacit in what follows.
C. The adélic reformulation. Consider a connected SD (G, X) with
G simply connected. By a result of Cartan, this means that G(R) is
connected, hence acts on X via Ad. Let Kf ≤ G(Af ) be a (“sufficiently
small”) compact open subgroup and (referring to Lemma 4.5) Γ =
G(Q) ∩ Kf the corresponding subgroup of G(Q). Replacing the earlier
notation we write
Sh◦Kf (G, X)
for the associated locally symmetric variety.
Proposition 4.8. The connected Shimura variety Sh◦Kf (G, X) (∼
= Γ\X)
is homeomorphic to
G(Q)\X × G(Af )/Kf ,
where the action defining the double quotient is
g.(ϕ̃, a).k := ( g.ϕ̃ , gak ).
|{z}
g ϕ̃g −1
Remark 4.9. If G(R) = Gad (R)+ , then the quotient X × G(Af )/Kf can
be written G(AQ )/K, where K = KR × Kf with KR ⊂ G(R) maximal
compact.
Sketch of Proof. First note that [ϕ̃] 7→ [(ϕ̃, 1)] gives a well-defined map
from Γ\X to the double-quotient, since
[ϕ̃] = [ϕ̃0 ] =⇒ ϕ̃0 = γ.ϕ̃ (γ ∈ Γ)
=⇒ (ϕ̃0 , 1) = γ.(ϕ̃, 1).γ −1 .
Now by assumption G is semisimple, simply connected, and of noncompact type. Lemma 4.2 implies that
G(Af ) = G(Q).Kf ,
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
32
so that for any (ϕ̃, a) ∈ X × G(Af ) we have
(ϕ̃, a) = (ϕ̃, gk) = g.(g −1 .ϕ̃, 1).k.
This implies that our map is surjective; for injectivity,
[(ϕ̃, 1)] = [(ϕ̃0 , 1)] =⇒ (ϕ̃0 , 1) = g.(ϕ̃, 1).k −1 = (g.ϕ̃, gk −1 )
=⇒ g = k ∈ Γ, [ϕ̃0 ] = [ϕ̃].
Finally, since Kf is open, G(Af )/Kf is discrete and the map
X → X × G(Af )/Kf
ϕ̃ 7→
(ϕ̃, [1])
is a homeomorphism, which continues to hold upon quotienting both
sides by a discrete torsion-free subgroup.
More generally, given a Shimura datum (G, X̃) and compact open
Kf ≤ G(Af ), we simply define the Shimura variety
ShKf (G, X̃) := G(Q)\X̃ × G(Af )/Kf ,
which will be a finite disjoint union of locally symmetric varieties. The
disconnectedness, at first, would seem to arise from two sources:
(a)
(b)
X̃ not connected (the effect of orbiting by G(R))
the failure of strong approximation for reductive groups
(which gives multiple connected components even when X̃
is connected).
The first part of the Theorem below says (a) doesn’t contribute: the
indexing of the components is “entirely arithmetic”. First, some
Notations 4.10.
• X denotes a connected component of X̃;
• G(R)+ is the preimage of Gad (R)+ in G(R), and G(Q)+ = G(Q) ∩
G(R)+ ;
• ν : G → T denotes the maximal abelian quotient (from §1.A), and
ν
the composition Z ,→ G T is an isogeny;
• T (R)† := Im(Z(R) → T (R));
• Y := T (Q)† \T (Q).
∼
=
Theorem 4.11. (i) G(Q)+ \X×G(Af )/Kf −→ G(Q)\X̃×G(Af )/Kf .
(ii) The map
G(Q)+ \X × G(Af )/Kf G(Q)+ \G(Af )/Kf =: C
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
33
“indexes” the connected components. If Gder is simply connected, we
have
∼
=
C → T (Q)† \T (Af )/ν(Kf ) ∼
= T (Q)\Y × T (Af )/ν(Kf ).
ν
Henceforth, “C” denotes a set of representatives in G(Af ).
(iii) ShKf (G, X̃) ∼
= qg∈C Γg \X, a finite union, where
Γg := gKf g −1 ∩ G(Q)+ .
That |C| < ∞ is by Lemma 4.1. The rest is in [Milne2005, sec. 5]
with two of the easier points (not requiring the hypothesis on Gder )
described in the following
Exercise 4.12. (a) The preimage of [1] ∈ C is Γ\X ∼
= Sh◦ der (Gder , X),
Kf
where Γ = Kf ∩ G(Q)+ , and Kfder ≤ Gder (Af ) is some subgroup cointaining Kf ∩ Gder (Af ).
(b) For γ ∈ Γg , (γ.ϕ̃, g) ≡ (ϕ̃, g) in ShKf (G, X̃).
The key point here is that (iii) is an analytic description of what
will turn out to be Gal(C/E)-conjugates of Γ\X, in analogy to the
formula for σ A[a] in the main theorem of CM. Here E is the reflex field
of (G, X̃), which will turn out to be the minimal field of definition of
this very clever disjoint union.
Remark 4.13. (a) The definitions of Shimura varieties here are due to
Deligne (in the late 1970s), who conjectured that they are fine moduli varieties for motives. The difficulty is in showing that the Hodge
structures they parametrize are motivic.
(b) There is a natural construction of vector bundles VKf (π) on ShKf (G, X̃),
holomorphic sections of which are (holomorphic) automorphic forms
of level Kf and type π. (Here, π is a representation of the parabolic
subgroup of G(C) stabilizing a point of the compact dual X̌.) Their
higher cohomology groups are called automorphic cohomology.
(c) The inverse system7 Sh(G, X̃) of all ShKf (G, X̃) is a reasonably
nice scheme on which G(Af ) operates, and so for example one could
i
study the representation of this on lim
−→H (ShKf (G, X̃), O(VKf (π))).
D. Examples.
I. 0-dimensional Shimura varieties. Let T be a Q-algebraic torus satisfying SV5-6, ϕ̃ : S → T a homomorphism of R-algebraic groups
satisfying SV4, and Kf ≤ T (Af ) be compact open. Then
ShKf (T, {ϕ̃}) = T (Q)\T (Af )/Kf
7suppressed
in these notes since not needed for canonical models; it uses (4.2).
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
34
is finite by Lemma 4.1. These varieties arise as C in the Theorem, and
also from CM Hodge structures, which by definition have a torus as
MT group.
II. Siegel modular variety. Begin with a Q-symplectic space (V, ψ)
— i.e., a vector space V /Q together with a nondegenerate alternating
form ψ : V × V → Q — and set G =
ψ(gu, gv) = χ(g)ψ(u, v) (∀u, v ∈ V )
GSp(V, ψ) := g ∈ GL(V ) .
for some χ(g) ∈ Q∗
Exercise 4.14. Check that χ : G → Gm defines a character of G.
Now consider the spaces
±
X := J ∈ Sp(V, ψ)(R)
2
J
=
−e
and
ψ(u, Jv) is ± -definite
of positive- and negative-definite symplectic complex structures on VR ,
and regard
X̃ := X + q X −
as a set of homomorphisms via ϕ̃(a + bi) := a + bJ (for a + bi ∈
C∗ = S(R)). Then G(R) acts transitively on X̃, and the datum (G, X̃)
satisfies SV1-6. For any compact open Kf the attached Shimura variety
is a Siegel modular variety.
Now consider the set


A an abelian variety/C,






±Q a polarization of H1 (A, Q),
(A, Q, η) ∼



η : VAf → Vf (A)(= H1 (A, Af )) an


isomorphism sending ψ 7→ a · Q (a ∈ Af ) 
MKf :=
,
∼
=
where an isomorphism of triples
(A, Q, η) −→
(A0 , Q0 , η 0 )
∼
=
0
is an isogeny f : A → A sending Q0 7→ q · Q (q ∈ Q∗ ) such that for
some k ∈ Kf the diagram
VAf
η
·k
/
VAf
η0
/
Vf (A)
f
Vf (A0 )
commutes. MKf is a moduli space for polarized abelian varieties with
Kf -level structure. Write ϕ̃A for the Hodge structure on H1 (A), and
choose an isomorphism α : H1 (A, Q) → V sending ψ to Q (up to Q∗ ).
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
35
Proposition 4.15. The (well-defined) map
MKf −→ ShKf (G, X̃)
induced by
(A, Q, η) 7−→ (α ◦ ϕ̃A ◦ α−1 , α ◦ η)
is a bijection.
III. Shimura varieties of PEL type. This time we take (V, ψ) to be a
symplectic (B, ∗)-module, i.e.
• (V, ψ) is a Q-symplectic space;
• (B, ∗) is a simple Q-algebra with positive involution ∗ (that is,
tr(B⊗Q R)/R (b∗ b) > 0); and
• V is a B-module and ψ(bu, v) = ψ(u, b∗ v).
We put G := AutB (V ) ∩ GSp(V, ψ), which is of generalized SL, Sp, or
SO type (related to the Albert classification) according to the structure of (BQ̄ , ∗). (Basically, G is cut out of GSp by fixing tensors in
T 1,1 V .) The (canonical) associated conjugacy class X̃ completes this
to a Shimura datum, and the associated Shimura varieties parametrize
Polarized abelian varieties with Endomorphism and Level structure
(essentially a union of quotients of MT domains cut out by a subalgebra E ⊆ End(V )). They include the Hilbert and Picard modular
varieties.
IV. Shimura varieties of Hodge type. This is a straightforward generalization of Example III, with G cut out of GSp(V, ψ) by fixing tensors
of all degrees.
Remark 4.16. In both Examples III and IV, X is a subdomain of a
Siegel domain, so “of Hodge type” excludes the type D and E Hermitian
symmetric domains which still do yield Shimura varieties parametrizing
equivalence classes of Hodge structures. So the last example is more
general still:
V. Mumford-Tate groups/domains with vanishing IPR. This was already partially dealt with in §4.B. In the notation from §1.D, let
G := Mt and
X̃ = Mt (R).ϕ̃ =: Dt
for some ϕ̃ with MT group Mt . (Note that X will be Dt+ .) Under
the assumption that Dt has trivial infinitesimal period relation, each
choice of level structure
Kf ≤ Mt (Af )
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
36
will produce Shimura varieties. These take the form
qg∈C Γg \Mtad (R)+ /KR (KR maximal compact)
with components parametrizing Γg -equivalence classes of higher weight
Hodge structures.
In addition to the examples in §1.D, one prototypical example is the
MT group and domain for HS of weight 3 and type (1, n, n, 1) with
endomorphisms by an imaginary quadratic field, in such a way that
the two eigenspaces are V 3,0 ⊕ V 2,1 and V 1,2 ⊕ V 3,0 . (In particular,
note that Mt (R) ∼
= GU (1, n).) This has vanishing IPR and yields a
Shimura variety.
Remark 4.17. (i) Even in the not-necessarily-Hermitian-symmetric case,
where Dt = M (R)/Hϕ̃ is a general MT domain, there is still a canonical
homeomorphism
(4.3)
∼
M (Q)\M (A)/Hϕ̃ × Kf
M (Q)\Dt × M (Af )/Kf
=
homeo
∼
=
qg∈C Γg \Dt+ =: GSKf (M, Dt ).
The right-hand object is a Griffiths-Schmid variety, which is in general only complex analytic. The construction of vector bundles VKf (π)
mentioned in Remark 4.13(b) extends to this setting. GSKf (M, Dt ) is
algebraic if and only if Dt+ fibers holomorphically or antiholomorphically over a Hermitian symmetric domain [GRT2013]. It is considered
to be a Shimura variety under the slightly more stringent condition
that the IPR be trivial.
(ii) Continuing in this more general setting, let α ∈ M (Q) and write
finite
Kf αKf = qi Kf ai for some ai ∈ M (Af ). This gives rise to an analytic
correspondence
X
[m∞ , mf ], [m∞ , mf a−1
i ]
[m∞ ,m ]∈M (A)
f
i
in the self-product of the LHS of (4.3). The endomorphism induced in
automorphic cohomology groups
Ai (π) := H i GSKf (M, Dt ), O(VKf (π))
is called the Hecke operator associated to α.
(iii) For simplicity, assume that the representations π are 1-dimensional
(so that the VKf (π) are line bundles) and that the Γg are co-compact.8
8Similar
remarks apply in the non-co-compact case, except that one has to restrict to a subgroup of cuspidal classes in Ai (π), whose behavior for i > 0 is not
yet well-understood.
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
37
In the Shimura variety case, the eigenvectors of Hecke operators in
A0 (π) are the arithmetically interesting automorphic forms. In the
non-algebraic Griffiths-Schmid case, provided π is “regular”, we have
A0 (π) = {0}; however, the automorphic cohomology Ai (π) will typically be nonzero for some i > 0. It doesn’t seem too far-fetched
to hope that the Hecke eigenvectors in this group might hold some
yet-to-be-discovered arithmetic significance. Recent work of Carayol
[Carayol2005] (in the non-co-compact case) seems particularly promising in this regard.
5. Fields of definition
Consider a period domain D for Hodge structures (on a fixed Qvector space V , polarized by a fixed bilinear form Q) with fixed Hodge
numbers. The compact dual of a Mumford-Tate domain DM = M (R).ϕ̃ ⊆
D is the M (C)-orbit of the attached filtration Fϕ̃• ,
ĎM = M (C).Fϕ̃• .
It is a connected component of the “MT Noether-Lefschetz locus” cut
out of Ď by the criterion of (a Hodge flag in Ď) having MT group
contained in M ,
ĎM ⊆ NˇLM ⊆ Ď.
Now, NˇLM is cut out by Q-tensors hence defined over Q, but its components9 are permuted by the action of Aut(C). The fixed field of the
subgroup of Aut(C) preserving ĎM , is considered its field of definition;
this is defined regardless of the vanishing of the IPR (or DM being Hermitian symmetric). What is interesting in the Shimura variety case, is
that this field has meaning “downstairs”, for Γ\DM — even though the
upstairs-downstairs correspondence is highly transcendental.
A. Reflex field of a Shimura datum. Let (G, X̃) be a Shimura
datum; we start by repeating the definition just alluded to in this
context. Recall that any ϕ̃ ∈ X̃ determines a complex cocharacter of
G by z 7→ ϕ̃C (z, 1) =: µϕ̃ (z). (Complex cocharacters are themselves
more general and essentially correspond to points of the compact dual.)
Write, for any subfield k ⊂ C,
C(k) := G(k)\Homk (Gm , Gk )
for the set of G(k)-conjugacy classes of k-cocharacters. The Galois
group Gal(k/Q) acts on C(k), since Gm and G are Q-algebraic groups.
9which
are M (C)-orbits, as M is (absolutely) connected; note that this does not
mean M (R) is connected.
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
38
The element
c(X̃) := [µϕ̃ ] ∈ C(C)
is independent of the choice of ϕ̃ ∈ X̃.
Definition 5.1. E(G, X̃) is the fixed field of the subgroup of Aut(C)
fixing c(X̃) as an element of C(C).
Examples 5.2. (a) Given A an abelian variety of CM type (E, Φ), E 0
the associated reflex field (cf. §3.A), ϕ̃ the Hodge structure on H 1 (A),
and
T = Mϕ̃ ⊆ ResE/Q Gm
the associated MT group. Then µϕ̃ (z) multiplies H 1,0 (A) (the Φeigenspaces for E) by z, and H 0,1 (A) (The Φ̄-eigenspaces for E) by
1. Clearly Ad(σ)µϕ̃ (z) (for σ ∈ Aut(C)) multiplies the σΦ-eigenspaces
by z, while T (C) acts trivially on HomC (Gm , TC ). Consequently, σ
fixes c({ϕ̃}) if and only if σ fixes Φ, and so
E(T, {ϕ̃}) = E 0 .
(b) For an inclusion (G0 , X̃ 0 ) ,→ (G, X̃), one has E(G0 , X̃ 0 ) ⊇ E(G, X̃).
Every X̃ has ϕ̃ factoring through rational tori, which are then CM
Hodge structures. Each torus arising in this manner splits over a CM
field, and so E(G, X̃) is always contained in a CM field. (In fact, it is
always either CM or totally real.)
(c) In the Siegel case, E(G, X̃)
= Q; while for a PEL Shimura datum,
E(G, X̃) = Q {tr (b|T0 A )}b∈B .
Let T be a Q-algebraic torus, µ a cocharacter defined over a finite
extension K/Q. Denote by
r(T, µ) : ResK/Q Gm → T
the homomorphism given on rational points by
K ∗ → T (Q)
Y
k→
7
φ(µ(k)).
φ∈Hom(K,Q̄)
As in Example 5.2(b), every (G, X̃) contains a CM-pair (T, {ϕ̃}). The
field of definition of µϕ̃ and the reflex field E(T, {ϕ̃}) are the same;
denote this by E(ϕ̃). The map
r(T, µϕ̃ ) : ResE(ϕ̃)/Q Gm → T
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
39
yields on AQ -points
A×
E(ϕ̃)
r(T,µϕ̃ )
/
T (AQ )
project
/
3
T (Af ).
=:rϕ̃
Example 5.3. In Example 5.2(a) above, we have E(ϕ̃) = E 0 ; assume
for simplicity that T = ResE/Q Gm . A nontrivial computation shows
that the r(T, µϕ̃ ) part of this map is the adelicized reflex norm
×
NΦ0 (AQ ) : A×
E 0 → AE .
B. Canonical models. The Shimura varieties we have been discussing
— i.e., ShKf (G, X̃) — are finite disjoint unions of locally symmetric
varieties, and hence algebraic varieties defined a priori over C. More
generally, if Y is any complex algebraic variety, and k ⊂ C is a subfield,
a model of Y over k is
• a variety Y0 over k, together with
θ
• an isomorphism Y0,C →
Y.
∼
=
For general algebraic varieties, it is not true that two models over
the same field k are necessarily isomorphic over that field. But if we
impose a condition on how Gal(C/E10 ) acts on a dense set of points on
any model, then the composite isomorphism
Y0,C
θ
∼
=
/
Y
θ̃
∼
=
/
#
f0,C
Y
f0 isomorphic
is forced to be Gal(C/E10 )-equivariant, making Y0 and Y
0
over E1 . Repeating this criterion for more point sets and number fields
f0 to be isomorphic over ∩i E 0 .
Ei0 , forces Y0 and Y
i
To produce the dense sets of points we need the following
Definition 5.4. (a) A point ϕ̃ ∈ X̃ is a CM point, if there exists a
(minimal, Q-algebraic) torus T ⊂ G such that ϕ̃(S(R)) ⊂ T(R).
(b) (T, {ϕ̃}) is then a CM pair in (G, X̃).
Remark 5.5. Such a ϕ̃ exists since in a Q-algebraic group every conjugacy class of maximal real tori contains one defined over Q. To get
density in X̃, look at the orbit G(Q).ϕ̃. To get density, more importantly, in ShKf (G, X̃), look at the set {[(ϕ̃, a)]}a∈G(Af ) .
To produce the condition on Galois action, recall for any CM field
E 0 the Artin reciprocity map
0 ab
0
.
artE 0 : A×
Gal
(E
)
/E
0
E
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
40
Definition 5.6. A model MKf (G, X̃) , θ of ShKf (G, X̃) over E(G, X̃)
is canonical, if for every
•
•
•
•
CM (T, ϕ̃) ⊂ (G, X̃)
a ∈ G(Af )
σ ∈ Gal E(ϕ̃)ab /E(ϕ̃)
×
s ∈ art−1
E(ϕ̃) (σ) ⊂ AE(ϕ̃) ,
θ−1 [(ϕ̃, a)] is a point defined over E(ϕ̃)ab , and
(5.1)
σ.θ−1 [(ϕ̃, a)] = θ−1 .[(ϕ̃, rϕ̃ (s)a)].
Remark 5.7. In (5.1), the rϕ̃ is essentially a reflex norm.
The uniqueness of the canonical model is clear from the argument
above — if one exists — since we can take the Ei0 to be various E(ϕ̃)
for CM ϕ̃, whose intersections are known to give E(G, X̃).
The existence of canonical models is known for all Shimura varieties
by work of Deligne, Shih, Milne and Borovoi. To see how it might come
about for Shimura varieties of Hodge type, first note that by
• Baily-Borel,10
Y := ShK (G, X̃) is a variety over C. Now we know that
• ShK is a moduli space for certain abelian varieties,
say A → Y. Let E = E(G, X̃) and σ ∈ Aut(C/E). Given P ∈ Y(C)
we have an equivalence class [AP ] of abelian varieties, and we define a
map
(σ Y)(C) → Y(C)
by
σ(P ) 7−→ [σ AP ].
That σ AP is still “in the family A” follows from
• the definition of the reflex field
and
• Deligne’s theorem (cf. [Deligne1982], or [?]) that the Hodge
tensors determining A are absolute.
That these maps produce regular (iso)morphisms
fσ : σ Y → Y
boils down to
• Borel’s theorem (§2).
10The
bullets in this paragraph distinguish the ingredients that go into the proof
of existence
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
41
Now Y has (for free) a model Y0 over some finitely generated extension
L of E, and using
• |Aut(Y)| < ∞ (cf. [Milne2005], Thm. 3.21)
we may deduce that for σ 0 fixing L
σ0
fσ 0
Y`
/
σ0 θ
?
Y
θ
Y0
commutes. At this point it makes sense to spread Y0 out over E — i.e.
take all Gal(C/E)-conjugates, viewed as a variety via
Y0
/
SpecL
3
/
SpecE.
The diagram
σ
Y
O
fσ
∼
=
/
YO
σθ
∼
=
∼
= θ
σ
Y0
Y0
shows that the spread is constant; extending it over a quasi-projective
base shows that Y has a model defined over a finite extension of E.
(To get all the way down to E requires some serious descent theory.)
Finally, that the action of Aut(C/E) on the resulting model implied
by the {fσ } satisfies (5.1) (hence yields a canonical model), is true by
• the main theorem of CM.
In fact, (5.1) is precisely encoding how Galois conjugation acts on various Ab(OE , Φ) together with the level structure.
So the three key points are:
(1)
the entire theory is used in the construction of canonical
models;
(2)
ShKf (G, X̃) is defined over E(G, X̃) independently of Kf ;
and
(3)
the field of definition of a connected component ShKf (G, X̃)+
is contained in E(G, X̃)ab and gets larger as Kf shrinks (and
the number of connected components increases).
C. Connected components and VHS. Assume Gder is simply connected. The action on CM points imposed by (5.1) turns out to force
the following action on
π0 ShKf (G, X̃) ∼
= T (Q)\Y × T (Af )/ν(Kf ),
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
42
ν
where G T is the maximal abelian quotient. For any ϕ̃ ∈ X̃, put
r = r(T, ν ◦ µϕ̃ ) : A×
→ T (AQ ) .
E(G,X̃)
ab
Then for σ ∈ Gal E(G, X̃) /E(G, X̃) and s ∈ art−1
(σ) one has
E(G,X̃)
σ.[y, a] = [r(s)∞ .y, r(s)f .a].
Assume for simplicity Y is trivial. Let E denote the field of definition
of the component S := ShKf (G, X̃)+ over [(1, 1)], which is a finite
abelian extension of E 0 := E(G, X̃). From the above description of the
Galois action, we get
E = fixed field of artE 0 rf−1 (T (Q).ν(Kf )) .
That is, by virtue of the theory of canonical models we can essentially
write down a minimal field of definition of the locally symmetric variety
S.
Example 5.8. Take the Shimura datum (G, X̃) = (T, {ϕ̃}) associated
to an abelian variety with CM by E, so that E 0 is the reflex field (and
E the field of definition of the point it lies over in a relevant Siegel
modular variety). Let Kf = UI for I ∈ I(E) and consider the diagram
A×
E 0 ,f
/
NΦ0 (=rf )
artE 0
Gal (E 0 )ab /E 0
/
NΦ0
//
A×
E,f
∼
= artE
artE
Gal E ab /E
E × \A×
E,f /UI
//
Gal (EI /E) ,
where NΦ0 exists (and is continuous) in such a way that the left-hand
square commutes. Writing “F F ” for fixed field, we get that
ab
×
= F F N−1
Gal(E
/E
)
.
E = F F artE 0 NΦ−1
0
0 (E UI )
I
Φ
In case the CM abelian variety is an elliptic curve, NΦ0 and NΦ0 are
essentially the identity (and E 0 = E), so
E = F F Gal(E ab /EI ) = EI .
It is a well-known result that, for example, the j-invariant of a CM elliptic curve generates (over the imaginary quadratic field E) its Hilbert
class field E(1) . We also see that the fields of definition of CM points in
the modular curve X(N ) are ray class fields modulo N .
We conclude by describing a possible application to variations of
Hodge structure. Let V → S be a VHS with reference Hodge structure
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
43
Vs over s ∈ S. The underlying local system V produces a monodromy
representation
ρ : π1 (S) → GL(Vs ),
and we denote ρ(π1 (S)) =: Γ0 with geometric monodromy group
Π := identity component of Q-Zariski closure of Γ0 .
Moreover, V has a MT group M ; and we make the following two crucial
assumptions:
• Π = M der ; and
• DM has vanishing IPR.
In particular, this means that the quotient of DM by a congruence
subgroup is a connected component of a Shimura variety, and that Π
is as big as it can be.
For any compact open Kf ⊆ M (Af ) such that Γ := Kf ∩M (Q) ⊇ Γ0 ,
V gives a period (analytic) mapping
an
+
an
∼
.
Ψan
Kf : SC → Γ\DM = ShKf (M, DM ) ⊗E C
This morphism is algebraic by Borel’s theorem, and has minimal field
of definition (trivially) bounded below by the field of definition E of
ShKf (M, DM )+ , which henceforth we shall denote E(Kf ).
The period mapping which gives the most information about V, is
the one attached to the smallest congruence subgroup Γ ⊂ M (Q) containing Γ0 . Taking then the largest Kf with Kf ∩ M (Q) equal to this
Γ, minimizes the resulting E(Kf ). It is this last field which it seems
natural to consider as the reflex field of a VHS — an “expected lower
bound” for the field of definition of a period mapping of V. Furtherπ
more, if V arises (motivically) from X → S, then assuming Deligne’s
absolute Hodge conjecture (cf. [Deligne1982, ?]), the Q̄-spread of π
produces a period mapping into DM modulo a larger Γ, and our “reflex
field of V” may be an upper bound for the minimal field of definition of
this period map.
At any rate, the relations between fields of definition of
• varieties Xs ,
• transcendental period points in DM , and
• equivalence classes of period points in Γ\DM ,
and hence between spreads of
• families of varieties,
• variations of Hodge structure, and
• period mappings,
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
44
is very rich. Our suggested definition may be just one useful tool, for
investigating the case where the period map is into a Shimura subdomain of a period domain.
References
[AMRT1975]
[BB1966]
[Borel1972]
[Borel1991]
[BS1973]
[BP2009]
[Carayol2005]
[Carlson]
[CCK1980]
[Cattani]
[CDK1995]
[CS]
[Deligne1979]
[Deligne1982]
[Green]
[GGK2010]
[GRT2013]
A. Ash, D. Mumford, M. Rapoport, and Y. Tai, “Smooth compactification of locally symmetric varieties”, Math. Sci. Press, Brookline,
Mass., 1975.
W. Baily and A. Borel, Compactification of arithmetic quotients of
bounded symmetric domains, Ann. of Math (2) 84, 442-528.
A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Diff. Geom. 6 (1972),
543-560.
———, “Linear algebraic groups”, Graduate Texts in Math. 126,
Springer-Verglas, New York, 1991.
A. Borel and J.-P. Serre, Corners and arithmetic groups, Comm.
Math. Helv. 4 (1973), 436-491.
P. Brosnan and G. Pearlstein, On the algebraicity of the zero locus
of an admissible normal function, preprint, arXiv:0910.0628.
H. Carayol, Cohomologie automorphe et compactifications partielles de certaines variétés de Griffiths-Schmid, Compos. Math.
141 (2005), 1081-1102.
J. Carlson, Period maps and period domains, this volume.
J. Carlson, E. Cattani, and A. Kaplan, Mixed Hodge structures and
compactifications of Siegel’s space, in “Algebraic Geometry Angers,
1979” (A. Beauville, Ed.), Sijthoff & Noordhoff, 1980, 77-105.
E. Cattani, Introduction to variations of Hodge structure, this volume.
E. Cattani, P. Deligne and A. Kaplan, On the locus of Hodge
classes, J. Amer. Math. Soc. 8 (1995), 483-506.
F. Charles and C. Schnell, Notes on absolute Hodge classes, this
volume.
P. Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modéles canoniques, in “Automorphic
forms, representations and L-functions”, Proc. Sympos. Pure Math.
33, AMS, Providence, RI, 1979, 247-289.
———, Hodge cycles on abelian varieties (Notes by J. S. Milne),
in “Hodge cycles, Motives, and Shimura varieties”, Lect. Notes in
Math. 900, Springer-Verlag, New York, 1982, 9-100.
M. Green, The spread philosophy in the study of algebraic cycles,
this volume.
M. Green, P. Griffiths, and M. Kerr, “Mumford-Tate groups and
domains: their geometry and arithmetic”, preprint, 2010, available
at http://www.math.wustl.edu/~matkerr/MTgroups.pdf.
P. Griffiths, C. Robles and D. Toledo, Quotients of non-classical
flag domains are not algebraic, preprint, 2013.
SHIMURA VARIETIES:
A HODGE-THEORETIC PERSPECTIVE
45
[Holzapfel1995] R.-P. Holzapfel, “The ball and some Hilbert problems”, Birkhäuser,
Boston, 1995.
[KU2009]
K. Kato and S. Usui, “Classifying spaces of degenerating polarized
Hodge structure”, Ann. Math. Stud. 169, Princeton Univ. Press,
Princeton, NJ, 2009.
[Kodaira1954] K. Kodaira, On Kähler varieties of restricted type (an intrinsic
characterization of algebraic varieties), Ann. of Math. 60 (1954),
28-48.
[Milne1990]
J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in “Automorphic forms, Shimura varieties,
and L-functions”, Perspect. Math. 10, Acad. Press, 1990, 283-414.
[Milne2005]
———, Introduction to Shimura varieties, in “Harmonic analysis,
the trace formula and Shimura varieties” (Arthur and Kottwitz,
Eds.), AMS, 2005.
[Milne2007]
———, The fundamental theorem of complex multiplication,
preprint, arXiv:0705.3446.
[Milne2010]
———, Shimura varieties and the work of Langlands, preprint,
available at http://www.jmilne.org.
[Mumford1969] D. Mumford, A note of Shimura’s paper “Discontinuous groups and
abelian varieties”, Math. Ann. 181 (1969), 345-351.
[Rotger2005]
V. Rotger, Shimura varieties and their canonical models: notes on
the course, Centre de Recerca Mathemática, Bellatera, Spain, Sept.
20-23, 2005.
[Serre1956]
J.-P. Serre, Géométrie algébrique et géométrie analytique, Annales
de l’Institut Fourier 6 (1956), 1-42.
[Silverman1995] J. Silverman, “Advanced topics in the arithmetic of elliptic curves”,
Grad. Texts in Math. 151, Springer-Verlag, New York, 1995.
Department of Mathematics, Campus Box 1146
Washington University in St. Louis
St. Louis, MO 63130
e-mail : [email protected]